arXiv subjects
Publications and source records attributed to Xi-Ping Zhu.
In this paper, we study the boundary regularity of harmonic map heat flows from a bounded domain in a metric measure space with generalized Ricci curvature bounded from below into a non-positively curved metric space.
We prove positive-time regularity for harmonic map heat flows from finite-dimensional $\mathrm{RCD}(K,N)$ spaces into complete $\mathrm{CAT}(0)$ spaces, without assuming that either the source or the target is smooth. For bounded-image initial data, the $\mathrm{EVI}$ gradient flow of the Dirichlet energy admits a representative that is locally Lipschitz jointly in space and time. Moreover, its spatial pointwise Lipschitz constant satisfies an Eells--Sampson-type parabolic Bochner inequality. The main difficulty is that the smooth parabolic perturbations used to select contact points are unavailable on an $\mathrm{RCD}$ source. We overcome it by a sliced contact-selection principle that converts the elliptic ABP estimate on $\mathrm{RCD}$ spaces into the space--time contact selection required by the Hamilton--Jacobi argument.
In this paper, we consider a vector-valued one-phase Bernoulli-type free boundary problem on a metric measure space $(X,d,μ)$ with Riemannian curvature-dimension condition $RCD(K,N)$. We first prove the existence and the local Lipschitz regularity of the solutions, provided that the space $X$ is non collapsed, i.e. $μ$ is the $N$-dimensional Hausdorff measure of $X$. And then we show that the free boundary of the solutions is an $(N-1)$-dimensional topological manifold away from a relatively closed subset of Hausdorff dimension $\leqslant N-3$.
In 1964, Eells and Sampson proved the celebrated long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. In 1992, Gromov and Schoen initiated the study of harmonic maps into $CAT(0) $ metric spaces. It naturally motivates the study of the harmonic map heat flow into singular metric spaces. In the 1990s, Mayer and Jost independently studied convex functionals on $CAT(0)$ spaces and extended Crandall-Liggett's theory of gradient flows from Banach spaces to $CAT(0) $ spaces to obtain the weak solutions for the harmonic map heat flow into $CAT(0)$ spaces. The weak solutions enjoy the favorable long-time existence, uniqueness and well-established long-time behaviors. It is a long-standing open question to ask if the weak solutions possess the Lipschitz regularity. Very recently, by using elliptic approximation method, Lin, Segatti, Sire, and Wang proved the weak solutions are Lipschitz in space and $1\over 2$-Hölder continuous in time, for a wide class of $CAT(0)$ spaces. In the present paper, we give a complete answer to the question. We show that every weak solution of the harmonic map heat flow into $CAT(0)$ spaces is Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality.
In this paper, we study the instability of naked singularities arising in the Einstein equations coupled with isothermal perfect fluid. We show that the spherically symmetric self-similar naked singularities of this system, are unstable to trapped surface formation, under $C^{1,α}$ perturbations of an external massless scalar field. We viewed this as a toy model in studying the instability of these naked singularities under gravitational perturbations in the original Einstein--Euler system which is non-spherically symmetric.
In 1983, Schoen-Uhlenbeck \cite{SU83} established boundary regularity for energy-minimizing maps between smooth manifolds with the Dirichlet boundary condition under the assumption that both the boundary and the data are of $C^{2,α}$. A natural problem is to study the qualitative boundary behavior of harmonic maps with rough boundary and/or non-smooth boundary data. For the special case where $u$ is a harmonic function on a domain $Ω\subset \mathbb R^n$, this problem has been extensively studied (see, for instance, the monograph \cite{Kenig94}, the proceedings of ICM 2010 \cite{Tor10} and the recent work of Mourgoglou-Tolsa \cite{MT24}). The $W^{1,p}$-regularity ($1<p<\infty$) has been well-established when $\partialΩ$ is Lipschitz (or even more general) and the boundary data belongs to $W^{1,p}(\partialΩ)$. However, for the endpoint case where the boundary data is Lipschitz continous, as demonstrated by Hardy-Littlewood's classical examples \cite{HL32}, the gradient $|\nabla u|(x)$ may have logarithmic growth as $x$ approaches the boundary $\partial Ω$ even if the boundary is smooth. In this paper, we first establish a version of the Gauss-Green formula for bounded domains in $RCD(K, N)$ metric measure space. We then apply it to obtain the optimal boundary regularity of harmonic maps from $RCD(K, N)$ metric measure spaces into $CAT(0)$ metric spaces. Our result is new even for harmonic functions on Lipschitz domains of Euclidean spaces.
In this work we are going to establish Hölder continuity of harmonic maps from an open set $Ω$ in an ${\rm RCD}(K,N)$ space valued into a ${\rm CAT}(κ)$ space, with the constraint that the image of $Ω$ via the map is contained in a sufficiently small ball in the target. Building on top of this regularity and assuming a local Lipschitz regularity of the map, we establish a weak version of the Bochner-Eells-Sampson inequality in such a non-smooth setting. Finally we study the boundary regularity of such maps.
We establish the boundary regularity of harmonic maps from $RCD(K, N)$ metric measure spaces into $CAT(0)$ metric spaces.
In this paper, we extend the classical Weyl's lemma to $RCD(K,N)$ metric measure spaces. As its applications, we show the local regularity of solutions for Poisson equations and a Liouville-type result for $L^1$ very weak harmonic functions on $RCD(K,N)$ spaces. Meanwhile, a byproduct is that we obtain a gradient estimate for solutions to a class of elliptic equations with dis-continuous coefficients.
In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.
In this paper, we will show the Yau's gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $κ$, $κ\geq0$, in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.
In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(K,N).$
In the previous paper \cite{L-Z}, for a characteristic problem with not necessarily small initial data given on a complete null cone decaying like that in the work \cite{Ch-K} of the stability of Minkowski spacetime by Christodoulou and Klainerman, we proved the local existence in retarded time, which means the solution to the vacuum Einstein equations exists in a uniform future neighborhood, while the global existence in retarded time is the weak cosmic censorship conjecture. In this paper, we prove that the local existence in retarded time still holds when the data is assumed to decay slower, like that in Bieri's work \cite{Bie} on the extension to the stability of Minkowski spacetime. Such decay guarantees the existence of the limit of the Hawking mass on the initial null cone, when approaching to infinity, in an optimal way.
We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future complete null cone with suitable decay, and show that the solution exists uniformly around the null cone for general such initial data. We can then define a segment of the future null infinity. The initial data are not required to be small and the decaying condition inherits from the works of \cite{Ch-K} and \cite{K-N}.
In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-manifolds with positive Yamabe invariants.
In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li-Yau's estimate for positve solutions of heat equations on Alexandrov spaces.
In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.